A Survey of Current Techniques for Reinforcement Learning

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A Survey of Current Techniques for

Reinforcement Learning

Magnus Borga

Tomas Carlsson

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Abstract

This survey considers response generating systems that improve their be-haviour using reinforcement learning. The di erence between unsupervised learning, supervised learning, and reinforcement learning is described. Two general problems concerning learning systems are presented; the credit as-signment problemand the problemofperceptual aliasing. Notations and some general issues concerning reinforcementlearning systems are presented. Rein-forcementlearning systems are further divided into two main classes;memory mapping and projective mapping systems. Each of these classes is described and some examples are presented. Some other approaches are mentioned that do not t into the two main classes. Finally some issues not covered by the surveyed articles are discussed, and some comments on the subject are made.

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Contents

1 Introduction

2

1.1 The Credit-Assignment Problem : : : : : : : : : : : : : : : : : 4

1.1.1 Temporal Di erence Methods : : : : : : : : : : : : : : 5

1.1.2 Dynamic Programming in Reinforcement Learning: : : 6

1.2 Perceptual Aliasing : : : : : : : : : : : : : : : : : : : : : : : : 7

2 Reinforcement Learning

10

2.1 Framework: : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11

2.2 Design : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13

3 Memory Mapping

16

3.1 The Pole-balancing Problem : : : : : : : : : : : : : : : : : : : 17

3.2 The Cerebellar Model Articulation Controller (CMAC) : : : : 19

4 Projective Mapping

21

4.1 Linear Systems : : : : : : : : : : : : : : : : : : : : : : : : : : 21

4.2 Nonlinear Projective Mapping : : : : : : : : : : : : : : : : : : 22

4.3 Back-propagation in RLS : : : : : : : : : : : : : : : : : : : : : 23

5 Other Approaches

26

5.1 The AI Approach : : : : : : : : : : : : : : : : : : : : : : : : : 26 5.2 Stochastic Automata : : : : : : : : : : : : : : : : : : : : : : : 26

6 Conclusions

28

1

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Chapter 1

Introduction

There is a wide class of systems that is calledresponse generating systems[6]. It could be any type of system that generates responses to certain stimuli, from a door bell that responds with a signal when a button is pressed, to a sophisticated robot or an animal that performs a complicated interaction with its environment. However complicated the system may be, it needs rules for how to produce proper responses to stimuli relevant to the task the system is to perform.

Principally there are two ways to instruct a system how to produce re-sponses. It can be done byprogrammingor bylearning. Programming means that the system is equipped with rules prede ning responses to all meaningful stimuli. A learning system, on the other hand, has no or little prede ned be-haviour. Instead it governs knowledge about what to do in di erent situations by trial and error. When dealing with complex problems the programmer's burden will become unbearable, since foreseeing every possible situation be-comes impossible. The learning system will of course also face a complex task, but building such a system need not be impossible. In the future, sys-tems may be placed in environments that humans have no experience from. Then it is obvious that learning systems are called for.

Learning is a broad concept and can span from the adaptive updating of lter coecients to the use of fragmented past experience to compile action strategies for new situations. This spectrum of possible interpretations is dicult to cover in one de nition of learning. It can be illustrated by two de nitions made by two mathematical learning theorists and two researchers in the eld of learning automata. The mathematiciansBush and Mosteller [4]

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consider any systematic change in behaviour as learning whether or not the change is adaptive, desirable for certain purposes or in accordance with any other such criterion. Narendra and Thathachar [15], two learning automata theorists, make the following de nition of learning: \Learning is de ned as any relatively permanent change in behaviour resulting from past experience, and a learning system is characterized by its ability to improve its behaviour with time, in some sense towards an ultimate goal".

Learning systems can be classi ed according to how the system is trained. Often the two groups unsupervised and supervised learning are suggested. Sometimes reinforcement learning is considered as a separate group to em-phasize the di erence between reinforcementlearning and supervised learning in general. The classes then become:

 unsupervised learning  supervised learning  reinforcement learning

In unsupervised learning there is no external unit or teacher to tell the system what is correct. The knowledge of how to behave is built into the system. This is clearly a limitation of the generality of the system. Most systems of this type are only used to learn ecient representations of signals. The learning methods are for example Hebbian learning [10] that performs feature mapping or principal component analysis, and competitive learning (winner-take-all) that perform clustering or pattern classi cation. Unsu-pervised learning systems can be very fast, since each component changes its behaviour simultaneously independently of the other components in the network. Some of these methods show similarities to some mechanisms in biological systems.

The opposite to unsupervised learning is supervised learning algorithms, where an external teacher must show the system the correct response to each input. The most used algorithm is back-propagation [16]. The problem with this method is that the correct answers to the di erent inputs have to be known, i.e. the problem has to be solved from the beginning, at least for some representative cases from which the system can generalize by interpola-tion. Another problem with this method is that it is very slow in multi-layer networks, since each layer has to wait for the error to be propagated from the

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last layer. Although the mechanisms used for learning in biological systems are mostly unknown, this type of learning systems seem to show very few similarities to biological systems.

Inreinforcement learning, however, the teacher tells the system how good or bad it performed but nothing about the desired responses. There are many problems where it is dicult or even impossible to tell the system which output is the desired for a given input (there could even be several correct outputs for each input) but where it is quite easy to decide when the system has succeeded or failed in a task (e.g. the pole balancing problem described in section 3.1).

There are two important problems all learning systems have in common; the credit-assignment problem and the problem of perceptual aliasing.

1.1 The Credit-Assignment Problem

In a complexsystemthat in some way is supposed to improveits performance, there is always a problem in deciding what part of the system that deserves credit or blame for the performance of the whole system. This is called the credit-assignment problem [13] or, to be more speci c, the structural

credit-assignment problem. In supervised learning the desired response is known and this problem can be solved, for instance with the back propagation algorithm in feed-forward neural nets [16].

The problem gets more complicated in reinforcement learning, where the information in the feedback to the system is limited, occurs infrequently or, as in many cases, a long time after the responsible actions have been taken. E.g. consider the loosing team in a football game, that scores one point during the last seconds of the game. It would not be clever to blame the last scored point. This kind of problem is called credit assignment over time or temporalcredit-assignment problem, investigated by Sutton in [19]. One solution to the temporal credit assignment problem in RLS is to supply the system with an internal reinforcement signal and let the system learn to improve its internal critic, the so calledadaptive critic method [3].

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loss win bad novel 90 % 10 %

Figure 1.1: An example to illustrate the advantage with TD-methods. See the text on page 5.

1.1.1 Temporal Di erence Methods

In [20] Sutton describes the methods of temporal di erences (TD). These methods enable a system to learn to predict the future behaviour of an in-completely known environment using past experience. TD methods can be used in systems where the input is a dynamic process. In these cases the TD methods take into account the sequential structure of the input, which the classical supervised learning methods do not.

Suppose that for each state s

k there is a value p

k that is an estimate of

the expected future result (e.g. the total accumulated reinforcement). In TD methods the value ofp

k depends on the value of p

k +1 and not only on the nal

result. This makes it possible for TD methods to improve their predictions during a process without having to wait for the nal result.

Let us look at an example. Consider a game, where a certain position has resulted in a loss in 90% of the cases and a win in 10% of the cases, see gure 1.1. This position is classi ed as a bad position. Now suppose that a player reaches a novel state (i.e. a state that has not been visited before) that inevitably leads to the bad state, and nally happens to lead to a win. If the player waits until the end of the game and only looks at the result, he would label the novel state as a good state, since it lead to a win. Most supervised learning methods would make such a conclusion. A TD method however, would classify the novel state as a bad state, since it leads to a bad state and the result probably will be a loss. It can make this classi cation without having to wait until the end of the game.

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Dynamic programming

Model (M) Utility function (U)

Secondary utility function (J)

Figure 1.2: Dynamic programming.

Sutton has proved a convergence theorem for one TD method1 that states

that the predictions for each state asymptotically converge to the maximum-likelihood predictions of the nal outcome for states generated in a Markov process, when presented to new data sequences. He has also proved that the predictions in this method converge to the maximum-likelihood estimates for such sequences under repeated training on the same set of data.

A TD method was rst used by A. L. Samuel in a checkers-playing pro-gramme in 1959 [17]. Most reinforcement learning methods could use TD methods, but TD methods could also be used in supervised learning sys-tems.

1.1.2 Dynamic Programming in

ReinforcementLearn-ing

There is a relationship between reinforcement learning and dynamic pro-gramming, as discussed by Whitehead in [26]. In [24] Werbos loosely de nes adaptive-critic methods as methods that try to approximate dynamic pro-gramming. Adaptive-critic methods are used to provide the RLS with an internal reinforcement signal. These systems have the capability to improve their performance during a trial (i.e. between the reinforcement signals).

Dynamic programming is the process of generating a secondary utility function (J) given a model (M) of the environment and a utility function

1In this TD method the prediction p

k only depends on the following prediction p

k +1

and not of later predictions. Other TD methods can take into account later predictions with a function that decreases exponentially with time.

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(U) in such a way that when J is optimized in the short term, U becomes

optimized in the long term,2 see gure 1.2. In adaptive-critic methods the

reinforcementsignal is the utility function (U) and the internal reinforcement

signal is the secondary utility function (J).

This relationship with dynamic programming has made more mathemat-ical treatment possible (e.g. an optimality theorem concerning one class of reinforcement learning algorithms [22]).

1.2 Perceptual Aliasing

A learning system perceives the external world through a sensory subsystem and represents the set of external states S

E with an internal state

represen-tation set S

I. This set can, however, rarely be identical to the real external

world state set S

E. Assuming a representation that completely describes

the external world in terms of objects, their features and relationships will be unrealistic even for relatively simple problem settings. Also, the internal state is inevitably limited by the sensor system, which leads to the fact that there is a many-to-many mapping between the internal and external states. That is, a state s

e 2 S

E in the external world can map into several internal

states and, what is worse, an internal state s i

2S

I could represent multiple

external world states. This phenomenon, illustrated in gure 1.3, is called

perceptual aliasing [25].

In the case when the learning system is an RLS, perceptual aliasing can cause the system to confound di erent external states that have the same internal state representation. This can cause the system to make wrong decisions. E.g. let the internal state s

i represent the external states  s a e and  s b e

and let the system take an action a. The expected reward for the decision to

take the action agiven state s, denoted (s i

;a), is now estimated by averaging

the rewards for that decision accumulated over time. If s a

e and  s b

e occur

approximately equally often and the actual accumulated reward for (s a e

;a)

is greater than the accumulated reward for (s b e

;a) then the expected reward

will be underestimated for (s a e

;a) and overestimated for (s b e

;a), leading to a

nonoptimal decision policy.

2Sometimes the whole optimization process is referred to as dynamic programming, i.e.

the optimization ofJ is included in the dynamic programming method.

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RLS S S s s E I i si si e se se 1 1 2 3 2 3

Figure 1.3: Perceptual aliasing.

There are cases when this is no problem and the phenomenon will be a feature instead. This happens if all decisions made by the RLS areconsistent. The reward for the decision (s

i

;a) then equals the reward for all

correspond-ing actual decisions (s k e

;a), where k is an index for this set of decisions. If

the mapping between the external and internal worlds is such that all deci-sions are consistent, then it is possible to collapse a large actual state space onto a small one where situations that are invariant to the task at hand are mapped onto one single situation in the representation space. In general this will seldom be the case and perceptual aliasing will be a problem.

In [25] Whitehead presents a solution to the problem of perceptual alias-ing for a restricted class of learnalias-ing situations. The basic idea is to detect inconsistent decisions by monitoring the estimated reward error, since the error will oscillate for inconsistent decisions, as discussed above. When an inconsistent decision is detected, the system is guided (e.g. by changing its direction of view) to another internal state, uniquely representing the desired external state. In this way all actions will produce consistent decisions, see gure 1.4. The guidance mechanisms are not learned by the system. This is noted by Whitehead who admits that a dilemma is left unresolved:

In order for the system to learn to solve a task, it must accurately represent the world with respect to the task. However, in order for the system to learn an accurate representation, it must know how to solve the task.

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1) 2) A B 1) 2) A B A B si s i si 1 2

Figure 1.4: Avoiding perceptual aliasing by observing the environment from another direction.

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Chapter 2

Reinforcement Learning

Reinforcement learning can be seen as a form of supervised learning, but there is one important di erence; while a supervised learning system is sup-plied with the desired responses under its training, the reinforcement learn-ing system is only supplied with some quality measure of the system's overall performance.

As an example, consider the procedure of training an animal. In general, there is no point in trying to explain to the animal how it should behave. The only way is simply to reward the animal when it does the right thing.

This, of course, makes a reinforcement learning system more general than a supervised learning system, since it can be trained on tasks where the exact desired responses for certain inputs are unknown, but where it is possible to obtain a qualitative measure of the systems performance. On the other hand some new problems are faced.

In [24] Werbos de nes an RLS as \any system that through interaction with its environment improves its performance by receiving feedback in the form of a scalar reward (or penalty) that is commensurate with the appro-priateness of the response". The goal for an RLS is simply to maximize this \reward", i.e. the accumulated value of the reinforcement signal r, see

gure 2.1. In this way,r can be said to de ne the problem to be solved.

In reinforcement learning the feedback to the system contains no gradient information, i.e. the system does not know in what direction to search for a better solution.1 Because of this, most RLS are supplied with some stochastic 1The problem can be solved by building a model of the environment that can be

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RLS r

s a

World

Critic

Figure 2.1: The Reinforcement Learning System in the external environment.

behaviour. This can be done by adding noise to the output as in [3]. Another way is by letting the output be generated from some probability distribution and moving the mean value to an optimal output and decreasing the variance as the output gets closer to the optimal value [9]. This stochastic behaviour can also help the system to avoid getting trapped in local minima.

This chapter rst presents the notations used in the report and then some issues that concerns reinforcement learning in general.

In this report, reinforcement learning systems (RLS) are divided into two main classes, according to how the mapping from input to output is made. The classes arememory mappingandprojective mappingand are discussed in chapter 3 and in chapter 4 respectively. Some other methods are mentioned in chapter 5.

2.1 Framework

The world, according to Whitehead [25], can be described by the tuple (S E ;A E ;W;R), where S E and A

E are the sets of world states and

physi-cal actions on the world respectively. The state transition function W maps

a state and an action into a new state, and the reward function R maps a

state into a real valued reward. A natural extension of this function is to let it map both the action and the state in which the action is taken into a

di erentiated with respect to the reinforcement signal, see section 2.2.

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P M W R B A I E C A E S ,R ,R SI

Figure 2.2: The subsystems in an RLS. P and M form the sensory-motor subsystem and B is the decision subsystem.

reward.

Again following Whitehead [25], the RLS can be divided into two subsys-tems; a sensory-motor subsystem and a decision subsystem, see gure 2.2. The purpose of the sensory-motor subsystem is threefold. It perceives the external world constituting a mapping from the set of external statesS

E and

the sensory-motor con guration C to the set of internal states S

I. On the

motor side the RLS has a set of internal motor commands A

I, that either

e ect the way the world is perceived by changing the con gurationC, or are

translated into external actionsA

E. The decision subsystem is the controller

and it has access to the internal state set S

I which it maps into the set of

internal actions A I.

The objective of the RLS is to learn and implement a decision policy that maximizes some functionf(r), that accumulates the received reward. Herer

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is the reinforcement, that is some measure of the overall performance of the system. This objective involves not only to implement an optimal mapping from S

I to A

I but also to control the systems sensory-motor subsystem in

order to gain knowledge about the external world and to avoid perceptual aliasing.

The reinforcement-signal can either be produced outside the RLS (e.g. by a supervisor that evaluates the performance of the system) or be produced inside the RLS (i.e. the rules of what is \good performance" are built into the system). These two methods could of course be combined. Some fundamental rules could be built into the system, while the task-dependent rules could be used by some external supervisor. In fact, there must always be a built-in rule that tells the RLS to maximizef(r).

2.2 Design

There are essentially three problems encountered when dealing with RLS:

 System architecture design  Construction of a critic

 Design of rules for improving the behaviour

Many system architecture designs have been proposed, but the main-stream one is the feed-forward input-output net. However, in [2], Ballard suggests that it is unreasonable to suppose that peripheral motor and sen-sory activity is correlated in a meaningful way. Instead, it is likely that abstract sensory and motor representations are built and related to each other. Also, combined sensory and motor information must be represented and used in the generation of new motor activity. This implies a learning hierarchy and that learning occurs on di erent temporal scales [5, 6, 7, 8]. Representing regularities in the sensory information that are relevant to the behaviour of the system as invariants gives the system more time to solve problems and makes it possible for the system to associate new situations with old experience.

The critic, or reinforcement, must be capable of evaluating the overall performance of the system and be informative enough to allow learning. The

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quality of the reinforcement signal is a critical factor, a ecting the perfor-mance of any RLS, since this signal is the one and only piece of information available to the system about its performance.

In some cases it is obvious how to choose the reinforcement signal. E.g. in the pole balancing problem described in section 3.1, the reinforcement signal is chosen as a negative value upon failure and zero otherwise [3]. Many times, however, it is not that clear how to measure the performance, and the choice of reinforcement signal could a ect the learning capabilities of the system.

The reinforcement signal should contain as much information as possible about the problem. The learning performance of a system could be improved considerable if a \pedagogical" reinforcement is used.

An example of this can be found in [9], where an RLS for learning real-valued functions is described. This system was supplied with two input variables and one output variable. In one case the system was trained on an XOR-task. Each input could be 0:1 or 0:9 and the output could be any

real number between 0 and 1. The optimal output values was 0:1 and 0:9

according to the logical XOR-rule. At rst the reinforcement signal was calculated as

r= 1,jer r or j;

where er r or is the di erence between the output and the optimal output.

The system some times converged to wrong results, and in several training runs it did not converge at all. A new reinforcement signal was calculated as

r 0= r+r task 2 : The term r

task was set to 0.5 if the latest outputs on similar inputs were

less than the latest outputs on dissimilar inputs and to -0.5 otherwise. With this reinforcement signal the system starts with trying to satisfy a weaker de nition of the XOR-task, which is that the output should be higher for dissimilar inputs than for similar inputs. The learning performance of the RLS improved in several ways with this new reinforcement signal.

Should the critic in general be built on the action taken, the new state or on the action taken given the state before the action? No unanimous answer to this seemingly simple question is to be found among learning researchers, illustrating the fact that a common theoretical framework is lacking in this area.

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Rules for improving the behaviourcan be based on essentiallytwo di erent strategies, either on di erentiating a model or on active exploration.

Di erentiating a model establishes the gradient of the reinforcement as a function of the model system parameters. The model can be known a priori and built into the system, or it can be learned and re ned during the training of the system. Knowing the gradient of the error means knowing in which direction in the parameter space to search for a better performance. How far to go in the direction of the gradient is, however, not known, and the step length must be short enough to prevent the system from oscillating. The update rules in systems incorporating projective mapping and model di erentiation often consist of adding a displacement to the weight vector in the direction of the gradient.

Active exploration is necessary when no model to help estimating the gradient is available. The only information is the reinforcement for the state-action pair. To be able to search the parameter space a stochastic behaviour component is needed. At rst the system may respond in a stochastic way, a behaviour that is modi ed as the system happens to do something that is rewarded. The modi cation can be the updating of an action probability distribution, as is often the case in memory mapping systems.

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Chapter 3

Memory Mapping

One approach to reinforcement learning is called memory mapping. This means that each input-vector is used as a pointer to a memory location that contains the response to that input, or contains a probability distribution from which the response is selected at random. In memory mapping the external state space must be quantized since the memory address is discrete and there is a nite number of memory locations. The quantization can be seen as a matter of system design, and be x for a given problem or learned by a system having a xed number of memory locations to play with [8]. The pure memory mapping approach would of course be impossible in a general system, since the memory size would become extremely large. However, in small specialized systems, e.g. in the pole-balancing problem mentioned below, this is no problem.

Extended stochastic learning automata [21] is a family name for one way of implementing the memory mapping concept. These automata make se-lections from a probabilistic and countable set of actions. The probabilities are then updated according to the evaluative feedback. An advantage of this method is that it can handle cases when two or more actions are equally good. One example is the control system by Musgrave and Loparo [14], pre-senting an RLS that learns by updating action probabilities P(a

j), where j

is an index for the set of possible actions.

Another related system is that of Whitehead and Ballard [25]. The sys-tem accomplish learning by updating an action value function Q(s;a) and

selecting the action having the largest action value for a given state. 16

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x F

Figure 3.1: The cart-pole system.

3.1 The Pole-balancing Problem

As an example of memory-mapping systems, consider a system designed by Barto, Sutton and Anderson that solves a control problem called pole balancing [3]. The system to be controlled consists of a cart on a track with an inverted pendulum (a pole) attached to it, as illustrated in gure 3.1. The cart can move between two xed endpoints of the track, and the pole can only move in the vertical plane of the cart and the track. The only way to control the movement of the cart is to apply a unit force F to it. The

force must be applied at every time-step, so the controller can only chose the direction of the force (i.e. right or left). This means that the output a from

the RLS is a scalar with only two possible values,F or,F. The state vector



s, that describes the cart-pole system, consists of four variables: x cart position,

 angle between the pole and the vertical,

_

x cart velocity, and

_

 angle velocity.

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ACE ASE s Cart-Pole system Decoder x1 x2 x n r r

Figure 3.2: ACE and ASE in the pole-balancing problem.

The reinforcementsignalris given to the RLS as a failure signal (i.e. negative

reinforcement) only when the pole falls or the cart hits the end of the track, otherwise r= 0. The task for the system is to avoid negative reinforcement

by choosing the \best" action a for each state s.

The four-dimensional state space is divided into 162 disjoint states by quantizing the state variables. Each of these states could be seen as a memory position that holds the \best" action for that state. The state vector s is

mapped by a decoder into an internal state vector x, in which one component

is 1 and the rest are 0. This vector is used as a pointer to the current state. This system consists of two parts: the ASE (associative search element) and the ACE (adaptive critic element), see gure 3.2. The ASE implements the mapping from the internal state vector x to the action a,

a(t) =f[ n X i=1 w i( t)x i( t) +noise(t)]

where f is the signum function and noise(t) is the mean zero Gaussian

dis-tribution. This mapping is changed by altering the weights w

i in such a way

that the internal reinforcementsignal ^ris maximized. The ACE produces the

internal reinforcement signal ^r as a mapping x7!^r. This mapping is chosen

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so that the accumulated value of the reinforcement signal is maximized. The ACE will develop a mapping that gives a positive value of ^r when the state

changes from a \dangerous" state to a state that is \safer" or vice versa. In this way the system can maximize the reinforcementr in the long term by

maximizing the internal reinforcement ^r for each decision. For more details,

see [3].

The ASE and ACE use weighted summation of the input vector and could therefore be used in the type of systems we call projective mapping if the decoder was removed.

3.2 The Cerebellar Model Articulation

Con-troller (CMAC)

One problem with the memory-mapping model, where each state of the en-vironment is represented by one memory location, is that the memory would grow to an enormous size as soon as the system get capabilities of any prac-tical use. Another problem is that the system has to be trained for every single state, i.e. there is no way for the system to generalize between states. One way of handling these problems in memory-mapping is called the

cerebellar model articulation controller (CMAC). This method was rst de-scribed by J. S. Albus in 1975 [1]. Instead of letting each state address one unique memory location only, this method allows each state to address a

set of memory locations. The contents of these locations are then summed together to form the output. This means that the system is capable of a some sort of generalization, i.e. a new stimulus that the system has not been exposed to before, but which is similar to some other stimuli that is known to the system, would generate a response that is similar to the response to the known stimulus. This method also reduces the amount of memory that is needed, since each state is represented by a combination of several memory locations. Consider a system with 100 memory locations where each state is represented by 10 locations. This would yield 100!=90! 610

19 states,

meaning that a large state space is mapped into a smaller memory space. The CMAC method has been used by C. Lin and H. Kim [12] in the same pole-balancing problem as described above. In this case the mapping between states and memory locations was done by hash coding. Due to interpolation

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between states, the CMAC system had a higher learning speed, and a smaller memory could be used.

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Chapter 4

Projective Mapping

The second main approach to reinforcement learning is projective mapping. The inner product between the input vector and the weight vector is calcu-lated, which means that this type of systems can handle continuous input vectors. The learning is done by changing the weights in such a way that the desired mapping is obtained. This is the most common method used in neural networks. Each component in the response vector is a function (linear or non-linear) of such an inner product.

These systems do not demand any unreasonable amounts of memory, and the possibility of generalization by interpolation is obvious.

This chapter begins with describing the special case of linear systems. In section 4.2 an example of nonlinear, and the use of back-propagation in reinforcement learning is discussed in section 4.3.

4.1 Linear Systems

Consider a multi-layerfeed-forward network with linear units, i.e. the output from each unit is calculated as

y i =  w T i  x

which means that the output vector from the rst layer is calculated as 

y=

W

x:

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The output ufrom the second layer is calculated in the same way,



u=

V

y;

which can be written like 

u=

VW

x=

Z

x:

This means that any multi-layer feed-forward net with linear units could be implemented as a single-layer net, i.e. as a multiplication of the input vector with a matrix.

4.2 Nonlinear Projective Mapping

Gullapalli [9] proposes an RLS that computes a real-valued output as a func-tion of a random activafunc-tion. The system consists of four parts:

 Parameter computers  Random number generator  Output function

 Update rules

The parameter computersis a number of functions g

j, that compute

dis-tribution parameters p

j and take the inner product of the input state vector



x and a weight vector w

j as their inputs: p j( t) =g j(  w j( t) T x(t))

These parameters are used by a random number generator, the distribution

d(p 1

;::;p

n), to produce random activation values a(t).

Finally the output is formed by feeding these activation values to an

output function f. This function is often a nonlinear squashing functionsuch

as e.g the logistic function:

f(x) = 1

1 +e ,x

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Modi cation of the weight vectors w

j for the case where the distribution

is the normal distribution d(p 1

;p 2) =

N(;); is accomplished by following

the update rules:

 w j( t+ 1) = w j( t) +  w( t)x(t)

where is a learning rate parameter and

w(

t) = (r(t),r^(t))

a(t),(t) (t)

denoting the reinforcementr(t) and the expected reinforcement

^

r(t) = v(t) T

x(t) +thr esh(t):

Here the vector v(t) has a function corresponding to that of w in the

calcu-lation of p 1(

t) =(t). Finallyp 2 =

 is a function of the expected

reinforce-ment so that a large expected reinforcereinforce-ment shows up as a small deviation

 and vice versa. Gullapalli motivates the use of the fraction in the

cal-culation of w(

t) by reasoning as follows. If the system has received more

reinforcement than expected, then it is desirable to move the mean closer to the current activation value by updating the mean in the direction of the fraction. On the other hand, if the reinforcement was lower than expected, the system should adjust its mean in the direction opposite to that of the fraction. Gullapalli claims that the above equations lead to such a behaviour.

4.3 Back-propagation in RLS

In [9] Gullapalli used w, the update vector of the output layer, as an error

estimate for back-propagation through a hidden layer. The system then becomes a hybrid between a reinforcement learning and supervised learning system, since the output layer faces a reinforcement learning task and the hidden layer faces a supervised learning task. He claims that  w under

certain conditions can be shown to be an unbiased estimate of the gradient of the reinforcement signal with respect to the activation in the hidden layer. Williams [28] uses back-propagation to update the system parameters in all layers. He assumes that the RLS output is computed from a probability distributiong i( z;w i ;x i) = Pfy i = zjw i ;x i g, where w i is a weight vector,  x i is

the input vector to the i:th unit andy

iis the output from the i:th unit. All the

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weight vectors of the network constitute a matrix denotedW. If @lng

i @w

ij exists

for all j, and the goal function to maximize is denoted J(W) = Efr jWg,

then a criterion for a learning algorithm to improve the value of J, at least

on average, is that the inner productEfWjWgrJ(W)0 with equality

only when rJ(W) = 0. Williams denotes this thehillclimbing property.

As mentioned earlier, it is necessary to build an internal model of the environment if back-propagation is to be used for establishing the gradient of the reinforcement signal. Two di erent kinds of information is then back-propagated, i.e. model prediction errors to train the internal model and predicted reinforcement to train the original network.

In [27] Williams introduced a class of RLS that update their weights on the form: w ij = ij( r,b ij) e ij

In back-propagation the direction to move in weight space is known but not how big the step in that direction should be. Thelearning rate factor

ij can

be considered as the step length and is supposed to be nonnegative and to satisfy some other conditions e.g to depend on the input x

i and to be small

enough to avoid overshots in the estimation process. Thereinforcement base-lineb ij is to be conditionally independent of y i, given W andx i. The adaptive

critic elementby Barto and Sutton [3] can be seen as adaptively updating this baseline or expected reinforcement. Lastly thecharacteristic eligibility,e

ij, is computed as e ij = @lng i @w

ij . The eligibility can be looked upon as a measure of

how receptive a weight is to credit or blame. A reinforcement algorithm on this form is shown to satisfy the previously mentioned hillclimbing property. This class of algorithms calls for back-propagation through the deterministic parts of the net to compute the eligibilitiese

ij.

Furthermore, if the probability distributions g

i are continuous and on the

form g = 1  h( y,  )

where and  are arbitrary translation and scaling parameters respectively,

then @J @y and y ,   @J

@y can be shown to be unbiased estimates of @J @ and

@J @

respectively. This result implies that it is possible to back-propagate through all parts of the net, deterministic as well as stochastic, compensating with a factor y ,

 when computing @J @.

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Williams also make some comments on how to make algorithms for ran-dom number generators having separate control over their parameters. Using the update rule mentioned before with the reinforcement baseline represent-ing the reinforcement in the past is said to lead to a reasonable learnrepresent-ing behaviour.

The problem of learning temporal behaviour is also treated by Williams in [27] where the system is supposed to work forktime steps and then receive

a reinforcementr. The update rule is extended also to incorporate time:

w ij = ij( r,b ij) k X t=1 e ij( t)

Now the eligibilitiese

ij not only depend on the input x

i but also on the time t. This extended learning algorithm is shown to possess the hillclimbing

property mentioned earlier.

In the same article Williams points out the exibility of the reinforcement baseline b

ij. This could be di erent for each unit in a network and be used

for individual tailoring of credit assignment. He suggests that informational connections could be added to a network to receive signals not a ecting the output but calculating the baseline of a unit.

Werbos [23] makes the noteworthy comment that these strategies leave out the crucial problem of maximizing some function f(r), of the

accumu-lated reinforcement. The established theorems talk about the expected next reinforcement Efr jWg instead of the expected accumulated reinforcement Eff(r)jWg, which is the more interesting quantity.

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Chapter 5

Other Approaches

Here some RLS designs are presented, that do not belong to the two previous described methods.

5.1 The AI Approach

Smith and Goldberg [18] have described an RLS that uses rules of type IF

conditionTHENactionto produce the output from the system. The rulebase is modi ed by a genetic algorithm.

Lee and Berenji [11] applied the AI approach with IF-THEN rules on the pole-balancing problem described earlier. This system was much faster in learning the pole-balancing task than Barto's method. It could also adapt to changes in length and mass of the pole without any failure.

5.2 Stochastic Automata

In their survey [15], Narendra and Thathachar describe a stochastic automa-ton. The automaton is equipped with a set of states, a set of actions and a corresponding set of action probabilities. An action is chosen at random using the action probabilities, and the reinforcement from the environment is observed. Now the probabilities are updated based on the received reinforce-ment and a new action is taken. Note that the actions taken are independent of the state of the environment. No sensory information but the

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ment signal is considered, which is the main di erence between stochastic automata and the extended stochastic automata mentioned earlier.

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Chapter 6

Conclusions

In most cases, the adaptive critic has been used to solve the temporal credit-assignment problem by learning to predict the expected reinforcement for the di erent states that the system's environment may enter. Then the assumption is made that some external states are better than others. For instance, in Barto's pole-balancing problem it is obvious which states are good and which are not; the \safe" states, where the cart is in the middle of the track and the pole stands straight up are of course better than more \dangerous" states like those at the end of the track. In the general case, however, things are not quite that easy. The expected reinforcement almost certainly depends not only on the state of the external world, but also on the action taken by the system. The internal reinforcement signal should therefore be a function of both the state vector and the action vector, i.e.

^

r=f(s ;a):

The generation of reinforcement signal is a subject that in most cases has not been discussed. Whether the reinforcement signal is generated inside the system or by some external unit it, is still an interesting issue how it should be determined. As we have seen, the way of generating reinforcement can be crucial to the systems learning capabilities. The reinforcement should give some hint of whether or not the system is \on the right track" in solving the problem, even if the best solution is far away. Perhaps the reinforcement signal could be adaptive, so that it initially gives a great deal of credit for a relative moderate improvement, but get harder in its critic as the system becomes better.

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The capabilities of generalization and association seem to be two impor-tant features in learning. Very few systems, however, have been investigated that use such features, and in most articles these capabilities are not even mentioned. Some times interpolation and extrapolation are described as gen-eralization, but useful de nitions of these two concepts are still wanted. The fact that a small distance between two state vectors only produces a small di erence between the generated action vectors is often used as an argument for seeing projective mapping systems as associating systems. This is, how-ever, not the only form of association. To be able to associate two parts of the curve outlined by the changing state vector may be even more important, since it re ects an ability to associate new situations with older ones. Once an association is made, it may be fruitful to test if it holds over a larger area by generalizing the correspondence to new areas and evaluate the hypothesis. A problem to be encountered before any associations or generalizations of successful actions and states are possible, is having the system generate good responses. In the case of memory mapping without gradient information, it is necessary for the system to respond in a random manner, waiting for an action to be rewarded. This may take a long time if the parameter space is large and has many dimensions. No solution to this problem is to nd among the references. For the class of projective mapping systems mentioned in [28], Williams has proved that they possess the hillclimbing property, meaning that the weight vectors are updated in a direction improving the expected reward. The measure is, however, a local one, and the problem of being trapped in a local extrema is not solved.

In this survey no practical applications of RLS have been found. No one seems to have been able to use these ideas in larger systems, e.g. in large multilayer neural networks. Unanimous design rules are lacking. Altogether, very little research has been done in this area compared to i.e. supervised learning. However, reinforcement learning seems to be a more general and natural way of learning than supervised learning and therefore better suited for autonomous learning systems.

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